How to Graph Linear Equations with a Restricted Domain on Both Sides

Graphing Linear Equations - How it Works - Video

Example 1

Example 1:

We have the equation y = -x - 2 for -1 ≤ x ≤ 1. The numbers after the word for are the limits on the equation or the restrictions that the equations has. We have to use those two numbers, -1 and 1, for our inputs. In order to make sure that we did everything correctly, we are going to find a third input. If we make a mistake, then the line will not go through all three points.

First we substitute -1 into the equation, y = -x - 2.

• y = -(-1) - 2 We substitute -1.

• y = 1 - 2 We multiply -1 * -1 ==> 1.

• y = -1 We subtract 1 - 2 ==> -1.

So our output value or value of y is -1. When we plot this point (-1, -1), the point is filled in because of the symbol in -1 ≤ x. Since it says less than or equal to, we have to fill in the point to show that we want the answer to be a part of the graph.

Next we substitute 0 into the equation, y = -x - 2.

• y = -(0) - 2 We substitute 0.

• y = 0 - 2 We multiply -1 * 0 ==> 0.

• y = -2 We subtract 0 - 2 ==> -2.

So our output value or value of y is -2. When we plot this point (0, -2), the point is filled in because the input value is in between the two limits.

Last we substitute 1 into the equation, y = -x - 2.

• y = -(1) - 2 We substitute 1.

• y = -1 - 2 We multiply -1 * 1 ==> -1.

• y = -3 We subtract -1 - 2 ==> -3.

So our output value or value of y is -3. When we plot this point (1, -3), the point is filled in because of the symbol in x ≤ 1. Since it says less than or equal to, we have to fill in the point to show that we want the answer to be a part of the graph.

Now we connect the points. The lines do not have arrows on either side because we have two limits, one on both side.

Example 2

Example 2:

We have the equation y = 2x - 3 for 0≤ x ≤ 4. The numbers after the word for are the limits on the equation or the restrictions that the equations has. We have to use those two numbers, 0 and 4, for our inputs. In order to make sure that we did everything correctly, we are going to find a third input. If we make a mistake, then the line will not go through all three points.

First we substitute 0 into the equation, y = 2x -3.

• y = 2(0) - 3 We substitute 0.

• y = 0 - 3 We multiply 2 * 0 ==> 0.

• y = -3 We subtract 0 - 3 ==> -3.

So our output value or value of y is -3. When we plot this point (0, -3), the point is filled in because of the symbol in 0 ≤ x. Since it says less than or equal to, we have to fill in the point to show that we want the answer to be a part of the graph.

Next we substitute 1 into the equation, y = 2x - 3.

• y = 2(1) - 3 We substitute 1.

• y = 2 - 3 We multiply 2 * 1 ==> 2.

• y = -1 We subtract 2 - 3 ==> -1.

So our output value or value of y is -1. When we plot this point (1, -1), the point is filled in because the input value is in between the two limits.

Last we substitute 4 into the equation, y = 2x - 3.

• y = 2(4) - 3 We substitute 4.

• y = 8 - 3 We multiply 2 * 4 ==> 8.

• y = 5 We subtract 8 - 3 ==> 5.

So our output value or value of y is 5. When we plot this point (4, 5), the point is not filled in because of the symbol in x < 4. Since it only says less than, we leave the point empty to show that we do not want the answer to be a part of the graph.

Now we connect the points. The lines do not have arrows on either side because we have two limits, one on both side.

Example 3

Example 3:

We have the equation y = (1/2)x - 1 for -2≤ x ≤ 4. The numbers after the word for are the limits on the equation or the restrictions that the equations has. We have to use those two numbers, 0 and 4, for our inputs. In order to make sure that we did everything correctly, we are going to find a third input. If we make a mistake, then the line will not go through all three points.

First we substitute -2 into the equation, y = (1/2)x - 1.

• y = (1/2)*(-2) - 1 We substitute -2.

• y = -1 - 1 We multiply (1/2) * (-2) ==> -1.

• y = -2 We subtract -1 - 1 ==> -2.

So our output value or value of y is -2. When we plot this point (-2, -2), the point is filled in because of the symbol in -2 < x. Since it only says less than, we leave the point empty to show that we do not want the answer to be a part of the graph.

Next we substitute 0 into the equation

• y = (1/2)*(0) - 1 We substitute -2.

• y = 0 - 1 We multiply (1/2) * (0) ==> 0.

• y = -1 We subtract 0 - 1 ==> -1.

So our output value or value of y is -1. When we plot this point (0, -1), the point is filled in because the input value is in between the two limits.

Last we substitute 4 into the equation

• y = (1/2)*(4) - 1 We substitute -2.

• y = 2 - 1 We multiply (1/2) * (4) ==> 2.

• y = 1 We subtract 2 - 1 ==> 1.

So our output value or value of y is 1. When we plot this point (4, 1), the point is not filled in because of the symbol in x ≤ 4. Since it says less than or equal to, we have to fill in the point to show that we want the answer to be a part of the graph.

Now we connect the points. The lines do not have arrows on either side because we have two limits, one on both side.

Example 4

Example 4:

We have the equation y = -2x - 1 for -1< x < 2. The numbers after the word for are the limits on the equation or the restrictions that the equations has. We have to use those two numbers, 0 and 4, for our inputs. In order to make sure that we did everything correctly, we are going to find a third input. If we make a mistake, then the line will not go through all three points.

First we substitute -1 into the equation, y = -2x - 1.

• y = -2*(-1) - 1 We substitute -1.

• y = 2 - 1 We multiply -2 * (-1) ==> 2.

• y = 1 We subtract 2 - 1 ==> 1.

So our output value or value of y is 1. When we plot this point (-1, 1), the point is filled in because of the symbol in -1 < x. Since it only says less than, we leave the point empty to show that we do not want the answer to be a part of the graph.

Next we substitute 1 into the equation, y = -2x -1.

• y = -2*(1) - 1 We substitute 1.

• y = -2 - 1 We multiply -2 * 1 ==> -2.

• y = -3 We subtract -2 - 1 ==> -3.

So our output value or value of y is -3. When we plot this point (1, -3), the point is filled in because the input value is in between the two limits.

Last we substitute 2 into the equation, y = -2x - 1.

• y = -2*(2) - 1 We substitute 1.

• y = -4 - 1 We multiply -2 * 2 ==> -4.

• y = -5 We subtract -4 - 1 ==> -5.

So our output value or value of y is 1. When we plot this point (2, -5), the point is not filled in because of the symbol in x < 2. Since it only says less than, we leave the point empty to show that we do not want the answer to be a part of the graph.

Now we connect the points. The lines do not have arrows on either side because we have two limits, one on both side.

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